Problem 14 - Algebra for College Students

Question

Verify that the points \((0,3),(2,-3)\), and \((-4,-5)\) are vertices of an isosceles triangle.

Step-by-Step Solution

Answer

The triangle is isosceles as AB = BC, both equal while CA is different.

Step 1: Calculate the Distance of AB

To find the distance between the points \(A(0, 3)\) and \(B(2, -3)\), we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substituting \(x_1 = 0\), \(y_1 = 3\), \(x_2 = 2\), and \(y_2 = -3\):\[ d = \sqrt{(2 - 0)^2 + (-3 - 3)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]

Step 2: Calculate the Distance of BC

To find the distance between the points \(B(2, -3)\) and \(C(-4, -5)\), we apply the same distance formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute \(x_1 = 2\), \(y_1 = -3\), \(x_2 = -4\), \(y_2 = -5\):\[ d = \sqrt{(-4 - 2)^2 + (-5 + 3)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

Step 3: Calculate the Distance of CA

To find the distance between the points \(C(-4, -5)\) and \(A(0, 3)\), using the distance formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute \(x_1 = -4\), \(y_1 = -5\), \(x_2 = 0\), \(y_2 = 3\):\[ d = \sqrt{(0 + 4)^2 + (3 + 5)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \]

Step 4: Compare the Distances

Upon calculating, we have the distances:- \(AB = 2\sqrt{10}\)- \(BC = 2\sqrt{10}\)- \(CA = 4\sqrt{5}\)Since \(AB = BC\) and both are different from \(CA\), the triangle formed is isosceles, having two sides of equal length.

Key Concepts

Distance Formula

The Distance Formula is a key concept in geometry that allows us to calculate the distance between two points on a Cartesian coordinate plane. The formula derives from the Pythagorean theorem and is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points.
• \(d\) is the distance between the points.

This formula allows us to measure how far apart two points are by using their coordinates. It is essential in verifying geometric properties like those of triangles by comparing the lengths of sides.
In the context of proving an isosceles triangle, the formula helps to find out if two sides of the triangle are equal, supporting triangle classification.

Vertices of a Triangle

Vertices of a triangle are the points where its sides intersect. In a two-dimensional plane, a triangle has three vertices. Each vertex is represented by its coordinate pair, such as \((x, y)\).
Identifying the vertices correctly is critical for calculating the distances between them using the Distance Formula.For example, in the problem, the triangle formed by the points \((0,3), (2,-3), (-4,-5)\) has:

• The vertex \(A\) at \((0,3)\)
• The vertex \(B\) at \((2,-3)\)
• The vertex \(C\) at \((-4,-5)\)

Understanding how to plot these vertices on a coordinate plane is crucial as it visually represents the triangle and helps in problem-solving.
This basic understanding builds a strong foundation for solving geometric problems, enabling you to apply formulas accurately and infer properties of the shapes formed.

Triangle Geometry

Triangle geometry is a pivotal branch of mathematics that deals with the properties and measurements of triangles. An isosceles triangle, a frequent subject of geometric study, is defined as a triangle with at least two sides of equal length.In this specific context, triangle geometry uses:

• Distance calculation for verifying equal sides.
• Properties like symmetry and angles can be explored once side lengths are established.

An essential approach in triangle geometry is to calculate the sides using the vertices' coordinates. Then, determine if the sides satisfy the conditions of being isosceles — two sides must be equal.
For the triangle with vertices \( (0,3), (2,-3), \) and \((-4,-5)\), the calculated distances were \(AB = 2\sqrt{10}\), \(BC = 2\sqrt{10}\), and \(CA = 4\sqrt{5}\), which confirm that \(AB = BC\). This shows the triangle is isosceles.Understandably, having two equal sides can lead to specific coordinated angles and potential symmetry features that further define the triangle's type.